Related works: classical approaches

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R ELIABILITY A LLOCATION : T HEORY

5.4. Related works: classical approaches

This section presents a systematic review of the most common RA methods available in literature.

5.4.1. Equal method

The simplest and easiest allocation method is the "Equal Reliability Allocation".

As it can be easily guessed from the name, this method allocates the same failure rate and the same reliability to all the components making up the

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system. This means that the weight factor πœ”πœ”π‘£π‘£ assessed using the Equal method is the same for all components i. As a consequence, the Equal method could be applied only to provide a first rough estimation of the reliability values to be allocated, but it cannot be considered a valuable solution.

The mathematical model of the equal allocation method is the following:

π‘…π‘…π‘£π‘£βˆ—(𝑑𝑑) = �𝑅𝑅𝑁𝑁 π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)= [π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)]𝑁𝑁1 (5.19) πœ”πœ”π‘£π‘£= 1

𝑅𝑅 (5.20)

5.4.2. ARINC method

The ARINC apportionment method was designed in 1964 by ARINC Research Corporation, a subsidiary of Aeronautical Radio, Inc [157].

This method is based on the assumption that the reliability of components can be assessed using previous calculations on similar components.

The mathematical expression of weight factors is the following:

πœ”πœ”π‘£π‘£= πœ†πœ†π‘£π‘£

πœ†πœ†π‘†π‘†π‘†π‘†π‘†π‘†= πœ†πœ†π‘£π‘£

βˆ‘π‘π‘π‘—π‘—=1πœ†πœ†π‘—π‘— (5.21)

Where πœ†πœ†π‘£π‘£ is the estimated failure rate of the component i-th obtained through a similar system and πœ†πœ†π‘†π‘†π‘†π‘†π‘†π‘† is the estimated failure rate of the whole architecture [157].

The peculiarity of the ARINC technique is that it is one of the few methods that considers historical failure data to assess the weight factors rather than quantitative influence factors like most of other techniques. As a matter of fact, ARINC requires the knowledge of past allocations on similar systems to allocate reliability to the various levels of the current system.

The main advantage of this method is essentially its simplicity of calculations which allows to rapidly implement the allocation. However, ARINC suffers many flaws, such as:

β€’ It is not possible to apply ARINC method to innovative systems since no past data related to a similar system are available.

β€’ All failure rates must be extracted from the same source (single database), as they must be comparable to each other in order to have an optimal allocation.

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5.4.3. AGREE method

AGREE (Advisory Group on Reliability of Electronic Equipment) technique considers three influence factors to calculate the weighting factors of each subsystem [158]. Complexity 𝐢𝐢𝑣𝑣 is assessed as the number of elements of the generic subsystem 𝑙𝑙𝑣𝑣 compared to the total number of components 𝑅𝑅𝑑𝑑𝑠𝑠𝑑𝑑 of overall configuration.

𝐢𝐢𝑣𝑣= 𝑙𝑙𝑣𝑣

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆 (5.22)

This technique also considers the importance 𝐼𝐼𝑣𝑣 of each subsystem i, where importance is defined as the probability that the system fails when the subsystem fails, thus πΌπΌπ‘£π‘£βˆˆ [0; 1] where 𝐼𝐼 = 1 stands for the most critical items, while 𝐼𝐼 = 0 means that the failure has no critical effects.

The third factor takes into account the effective time of use 𝑑𝑑𝑣𝑣′ of the subsystems, as follow:

𝑑𝑑𝑣𝑣′= 𝑑𝑑

𝑑𝑑𝑣𝑣 (5.23)

where 𝑑𝑑𝑣𝑣 is the time of use of item I, while 𝑑𝑑 is the time of use of the whole system.

According to the AGREE method [158], the reliability of a series architecture composed by N subsystems is defined as follow:

π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑) = οΏ½{1 βˆ’ 𝐼𝐼𝑣𝑣[1 βˆ’ π‘…π‘…π‘£π‘£βˆ—(𝑑𝑑𝑣𝑣)]}

𝑁𝑁 𝑣𝑣=1

= οΏ½οΏ½1 βˆ’ 𝐼𝐼𝑣𝑣�1 βˆ’ π‘˜π‘˜βˆ’πœ†πœ†π‘–π‘–βˆ—π‘‘π‘‘π‘–π‘–οΏ½οΏ½

𝑁𝑁 𝑣𝑣=1

(5.24)

Using the Taylor approximation of the exponential function π‘˜π‘˜βˆ’π‘₯π‘₯ β‰ˆ 1 βˆ’ π‘₯π‘₯ when π‘₯π‘₯ β†’ 0, then:

π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑) β‰ˆ οΏ½{1 βˆ’ 𝐼𝐼𝑣𝑣[1 βˆ’ (1 βˆ’ πœ†πœ†π‘£π‘£βˆ—π‘‘π‘‘π‘£π‘£)]}

𝑁𝑁 𝑣𝑣=1

= οΏ½{1 βˆ’ πΌπΌπ‘£π‘£πœ†πœ†π‘£π‘£βˆ—π‘‘π‘‘π‘£π‘£}

𝑁𝑁 𝑣𝑣=1

(5.25)

Introducing the Taylor approximation once again and rewriting the system reliability as exponential function:

π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑) β‰ˆ οΏ½οΏ½π‘˜π‘˜βˆ’πΌπΌπ‘–π‘–πœ†πœ†π‘–π‘–βˆ—π‘‘π‘‘π‘–π‘– οΏ½

𝑁𝑁 𝑣𝑣=1

= π‘˜π‘˜βˆ’ βˆ‘π‘π‘π‘†π‘†π‘†π‘†π‘†π‘†π‘–π‘–=1 οΏ½πΌπΌπ‘–π‘–πœ†πœ†π‘–π‘–βˆ—π‘‘π‘‘π‘–π‘–οΏ½ (5.26)

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To solve equation (5.26) it is necessary to rewrite the left term using the exponential function.

Thus, considering the properties of exponential and logarithmic functions, equation (5.26) can be rewritten as follow:

π‘˜π‘˜π‘ π‘ π‘›π‘›[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)]= π‘˜π‘˜βˆ’ βˆ‘π‘π‘π‘–π‘–=1οΏ½πΌπΌπ‘–π‘–πœ†πœ†π‘–π‘–βˆ—π‘‘π‘‘π‘–π‘–οΏ½ (5.27) 𝑐𝑐𝑙𝑙[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)] = βˆ’ οΏ½ πΌπΌπ‘£π‘£πœ†πœ†π‘£π‘£βˆ—π‘‘π‘‘π‘£π‘£

𝑁𝑁 𝑣𝑣=1

(5.28)

Multiplying and dividing the first term of equation (5.28) by the same quantity 𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆:

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆 𝑐𝑐𝑙𝑙[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)] = βˆ’ οΏ½ πΌπΌπ‘£π‘£πœ†πœ†π‘£π‘£βˆ—π‘‘π‘‘π‘£π‘£

𝑁𝑁 𝑣𝑣=1

(5.29)

However, 𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆 is the total number of components that make up the entire system. Thus, considering the definition of Complexity introduced by the AGREE method in equation (5.22), 𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆 can be rewritten as follow:

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆= οΏ½ 𝑙𝑙𝑣𝑣

𝑁𝑁 𝑣𝑣=1

(5.30)

Introducing equation (5.30) within equation (5.29):

οΏ½ πΌπΌπ‘£π‘£πœ†πœ†π‘£π‘£βˆ—π‘‘π‘‘π‘£π‘£ 𝑁𝑁 𝑣𝑣=1

= βˆ’π‘π‘π‘™π‘™[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)] βˆ™ βˆ‘ 𝑙𝑙𝑁𝑁 𝑣𝑣 𝑣𝑣=1

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆 (5.31)

Then, using the properties of the summation:

οΏ½ πΌπΌπ‘£π‘£πœ†πœ†π‘£π‘£βˆ—π‘‘π‘‘π‘£π‘£ 𝑁𝑁 𝑣𝑣=1

= βˆ’ οΏ½ οΏ½π‘™π‘™π‘£π‘£βˆ™π‘π‘π‘™π‘™[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)]

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆 οΏ½

𝑁𝑁 𝑣𝑣=1

(5.32)

πΌπΌπ‘£π‘£πœ†πœ†π‘£π‘£βˆ—π‘‘π‘‘π‘£π‘£= βˆ’π‘™π‘™π‘£π‘£βˆ™π‘π‘π‘™π‘™[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)]

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆 (5.33)

91 Introducing the definition of Complexity 𝐢𝐢𝑣𝑣 as in equation (5.22) and the definition of effective time as in equation (5.23) the latter became:

πœ†πœ†π‘£π‘£βˆ—= βˆ’πΆπΆπ‘£π‘£βˆ™ π‘‘π‘‘π‘£π‘£β€²βˆ™ 𝑐𝑐𝑙𝑙[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)]

𝐼𝐼𝑣𝑣𝑑𝑑 (5.34)

Now it is possible to define the weight factor of the AGREE method as a function of complexity, importance and effective time:

πœ”πœ”π‘£π‘£=πΆπΆπ‘£π‘£βˆ™ 𝑑𝑑𝑣𝑣′

𝐼𝐼𝑣𝑣 (5.35)

Introducing equation (5.35) and equation (5.2) within equation (5.34) the allocated failure rate according to the AGREE method could be expressed as:

πœ†πœ†π‘£π‘£βˆ—= βˆ’πœ”πœ”π‘£π‘£βˆ™ 𝑐𝑐𝑙𝑙[π‘…π‘…π‘†π‘†π‘†π‘†π‘†π‘†βˆ—(𝑑𝑑)]

𝑑𝑑 = πœ”πœ”π‘£π‘£βˆ™ πœ†πœ†π‘†π‘†π‘†π‘†π‘†π‘†βˆ— (5.36)

The AGREE technique is a milestone in RA approaches. However, it suffers major drawbacks, such as:

β€’ The importance factor, as it is defined, does not take into account the consequences that a subsystem failure induced on the system.

β€’ It requires Taylor approximation, thus obtaining approximate result.

β€’ The assessment of the weight factor takes into account only three influence factors.

5.4.4. FOO method

The FOO (Feasibility-Of-Objectives) technique was first introduced in 1976 by Anderson [159] and then included into the MIL-HDBK-338B Electronic Reliability Design Handbook from Department of Defense of USA in 1988 [148]

as a method to develop and implement reliability programs for generic military products. Following the FOO method, the subsystem allocation factors are computed as a function of four influence factors, namely complexity C, environmental factor E, state of the art A and operative time O. Each rank is estimated using both design engineering and expert judgments and it is based on a scale from 1 to 10 as detailed described in TABLE V.II.

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TABLE V.II

RULES FOR THE ASSESSMENT OF INFLUENCE FACTORS ACCORDING TO FOO METHOD. INFLUENCE FACTORS RATING

COMPLEXITY -C 12345678910 LOW MAX ENVIRONMENT CONDITION -E 12345678910

LOW MAX STATE OF THE ART -A 12345678910

MAX LOW OPERATING TIME -T 12345678910

MAX LOW

The rating values are then multiplied to achieve a partial weight factor 𝛽𝛽𝑣𝑣.

𝛽𝛽𝑣𝑣= πΆπΆπ‘£π‘£βˆ™ πΈπΈπ‘£π‘£βˆ™ π΄π΄π‘£π‘£βˆ™ 𝑂𝑂𝑣𝑣 (5.37) The final product has values ranging from 1 to 10000 and the subsystem ratings are normalized so that their sum is equal to 1.

Thus, the weighting factors are given by:

πœ”πœ”π‘£π‘£= πΆπΆπ‘£π‘£βˆ™ πΈπΈπ‘£π‘£βˆ™ π΄π΄π‘£π‘£βˆ™ 𝑂𝑂𝑣𝑣

βˆ‘ �𝐢𝐢𝑁𝑁𝑗𝑗=1 π‘—π‘—βˆ™ πΈπΈπ‘—π‘—βˆ™ π΄π΄π‘—π‘—βˆ™ 𝑂𝑂𝑗𝑗� =βˆ‘π›½π›½π‘£π‘£π›½π›½

𝑁𝑁 𝑣𝑣

𝑗𝑗=1 (5.38)

The FOO method is a simple technique easily implementable using software tools. However, it is characterized by some major flaws (quite similar to the RPN drawbacks described in Section 3.2):

β€’ The partial weight factor 𝛽𝛽𝑣𝑣 is not unique. In fact, different combinations of the influence factors could provide the same 𝛽𝛽𝑣𝑣.

β€’ Although the partial weight factor 𝛽𝛽𝑣𝑣 could assume values between 1 and 10000, there are many gaps in the range and only a very limited part of these 10000 possible values is obtained from a unique combination of factors.

β€’ All the different combinations of influence factors that lead to the same partial weight factor will also lead to the same allocated reliability. This may not be correct as the nature of the influence factors producing the same 𝛽𝛽𝑣𝑣 can be remarkably different.

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β€’ All the four influence factors have the same importance within the equation.

β€’ High subjectivity of the definition, which is deeply influenced by the expert’s judgments.

5.4.5. Bracha method

Bracha method uses the same factors of FOO technique (see TABLE V.II) but it privileges the state of the art factor A in the formula to calculate the partial weight factors 𝛽𝛽𝑣𝑣 [160]:

𝛽𝛽𝑣𝑣= π΄π΄π‘£π‘£βˆ™ (𝐢𝐢𝑣𝑣+ 𝑂𝑂𝑣𝑣+ 𝐸𝐸𝑣𝑣) (5.39) According to the Bracha method, the values of the influence factors are not determined by an expert like the FOO approach. Instead, the influence factor ratings are calculated through a set of complex mathematical models using several base factors, some of them are listed below:

β€’ the number of components of each subsystem;

β€’ the number of components of the most complex subsystem;

β€’ the number of redundancies;

β€’ the time of use of each subunit;

β€’ the operating time of each subsystem;

β€’ the applied stress;

β€’ the age of the database;

β€’ the time required to design the system.

These models result in a set of four influence factors mathematically estimated varying in the range from 0 to 1. The subsystem ratings are then normalized, therefore the weighting factors are given by [160]:

πœ”πœ”π‘£π‘£= π΄π΄π‘£π‘£βˆ™ (𝐢𝐢𝑣𝑣+ 𝑂𝑂𝑣𝑣+ 𝐸𝐸𝑣𝑣)

βˆ‘ [𝐴𝐴𝑁𝑁𝑗𝑗=1 π‘£π‘£βˆ™ (𝐢𝐢𝑣𝑣+ 𝑂𝑂𝑣𝑣+ 𝐸𝐸𝑣𝑣)] =βˆ‘π›½π›½π‘£π‘£π›½π›½

𝑁𝑁 𝑣𝑣

𝑗𝑗=1 (5.40)

The Bracha method is able to solve two out of five drawbacks of the FOO method, namely the high subjectivity of the factor definition and the same importance assigned to all the factor in the equation to calculate πœ”πœ”π‘£π‘£. However, it is not able to solve the other three major drawbacks of the FOO method, and it is also characterized by a high computational complexity due to the

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models required to assess the complexity C, the environmental factor E, the state of the art A and the operative time O.

5.4.6. Karmiol method

The Karmiol method is based on the assessment of four influence factors, namely complexity C, state of the art A, operative time O and Criticality K.

Each rank is estimated using both design engineering and expert judgments and it is based on a scale from 1 to 10 [161].

The procedure used to calculate the partial weight factor 𝛽𝛽𝑣𝑣 and the weight factor πœ”πœ”π‘£π‘£ is quite similar to the FOO model. The only difference is that the Karmiol method allows two different approaches. In the first one the partial weight factor 𝛽𝛽𝑣𝑣 is based on the product of the indexes similarly to the FOO, as follow:

𝛽𝛽𝑣𝑣= πΆπΆπ‘£π‘£βˆ™ π΄π΄π‘£π‘£βˆ™ π‘‚π‘‚π‘£π‘£βˆ™ 𝐾𝐾𝑣𝑣 (5.41) Then, the weight factor is achieved after a normalization process to ensure that equation (5.10) is satisfied. Thus:

πœ”πœ”π‘£π‘£= πΆπΆπ‘£π‘£βˆ™ π΄π΄π‘£π‘£βˆ™ π‘‚π‘‚π‘£π‘£βˆ™ 𝐾𝐾𝑣𝑣

βˆ‘ (𝐢𝐢𝑁𝑁𝑗𝑗=1 π‘£π‘£βˆ™ π΄π΄π‘£π‘£βˆ™ π‘‚π‘‚π‘£π‘£βˆ™ 𝐾𝐾𝑣𝑣 ) =βˆ‘π›½π›½π‘£π‘£π›½π›½

𝑁𝑁 𝑣𝑣

𝑗𝑗=1 (5.42)

Alternatively, it is possible to calculate the partial weight factor as sum of the indexes and then evaluate the weight factor after the normalization process, as follow:

𝛽𝛽𝑣𝑣= 𝐢𝐢𝑣𝑣+ 𝐴𝐴𝑣𝑣+ 𝑂𝑂𝑣𝑣+ 𝐾𝐾𝑣𝑣 (5.43) πœ”πœ”π‘£π‘£= 𝐢𝐢𝑣𝑣+ 𝐴𝐴𝑣𝑣+ 𝑂𝑂𝑣𝑣+ 𝐾𝐾𝑣𝑣

βˆ‘ (𝐢𝐢𝑁𝑁𝑗𝑗=1 𝑣𝑣+ 𝐴𝐴𝑣𝑣+ 𝑂𝑂𝑣𝑣+ 𝐾𝐾𝑣𝑣 ) =βˆ‘π›½π›½π‘£π‘£π›½π›½

𝑁𝑁 𝑣𝑣

𝑗𝑗=1 (5.44)

5.4.7. AWM method

In 1999 Kuo [162] introduced an Averaging Weighted Method (AWM) as a guide for reliability allocation design.

The method uses a questionnaire investigation to select the most influential system reliability factors such as complexity, state-of-the-art, system criticality, environment, safety, and maintenance in order to determine the subsystem reliability allocation ratings. All the influence factors included in Fig. 5.3 are

95 allowed, depending on the results of the questionnaire. Each rank is estimated on a scale from 1 to 10 using design engineering and expert judgments to obtain the subsystem reliability rate [162]. TABLE V.III shows the admissible influence factors and their rating rules according to the guidelines described in section 5.3.2.

TABLE V.III

INFLUENCE FACTORS ADMISSIBLE BY AWM ALLOCATION METHOD

INFLUENCE

FACTORS DESCRIPTION RATING

COMPLEXITY -C Number of components;

system architecture. 12345678910 LOW MAX ENVIRONMENT

CONDITION -E

External stress factors (humidity, temperature, vibration, etc.).

12345678910 LOW MAX STATE OF THE ART

-A

Scientific development in the system specific engineering context.

12345678910 MAX LOW CRITICALITY -K Subsystem importance;

consequences of a potential fault on the entire system.

12345678910 MAX LOW MAINTAINABILITY -M Average repair cost; average

repair time. 12345678910 LOW MAX SAFETY -R Impact of failure on system

safety 12345678910

MAX LOW

Considering a system composed by N subsystem, m is the number of influence factors and p the number of experts. Let π‘Œπ‘Œπ‘£π‘£π‘—π‘— denotes the j-th rating for subsystem i. 𝑋𝑋𝐾𝐾𝑣𝑣𝑗𝑗 is the j-th rating for subsystem i set by L-th expert and each factor is defined as follows:

π‘Œπ‘Œπ‘£π‘£π‘—π‘— =1

π‘˜π‘˜οΏ½ 𝑋𝑋𝐾𝐾𝑣𝑣𝑗𝑗

𝑝𝑝 𝑃𝑃=1

βˆ€π‘–π‘– = 1, … , π‘šπ‘šβˆ€π‘—π‘— = 1, … , 𝑅𝑅 (5.45)

Then, similarly to the Karmiol method, also in this case two different models can be used to allocate weighting factors πœ”πœ”π‘£π‘£.

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The geometric model is based on the product of the influence factors, and thus the weight factor is given by:

πœ”πœ”π‘£π‘£= βˆπ‘π‘π‘—π‘—=1π‘Œπ‘Œπ‘£π‘£π‘—π‘—

βˆ‘ βˆπ‘π‘ π‘Œπ‘Œπ‘£π‘£π‘—π‘— π‘šπ‘š 𝑗𝑗=1

𝑓𝑓=1 = 𝛽𝛽𝑓𝑓

βˆ‘π‘šπ‘š 𝛽𝛽𝑓𝑓

𝑓𝑓=1 (5.46)

While the arithmetic model is based on the sum of the influence factors:

πœ”πœ”π‘£π‘£= βˆ‘π‘π‘ π‘Œπ‘Œπ‘£π‘£π‘—π‘— 𝑗𝑗=1

βˆ‘ βˆ‘π‘π‘ π‘Œπ‘Œπ‘£π‘£π‘—π‘— π‘šπ‘š 𝑗𝑗=1

𝑓𝑓=1 = 𝛽𝛽𝑓𝑓

βˆ‘π‘šπ‘šπ‘“π‘“=1𝛽𝛽𝑓𝑓 (5.47)

The AWM has many advantages, such as:

β€’ The higher the experts number, the lower the impact of a possible evaluation error.

β€’ Minimum subjectivity issue due to factors assessment performed after a questionnaire-based investigation.

β€’ Possibility to choose which influence factors represent the best alternative to fit the specific system judging on system features. Thus, only the factors that actually influence the system performances are taken into consideration.

β€’ Low complexity.

The main drawback of the AWM method is the equal weight that the influence factors have in the final equations (5.46) and (5.47).

In document To my family, (Page 123-132)